Graduate Volume Element for Isotropic Harmonic oscillator

[Diracobama2181]

I am currently having trouble deriving the volume element for the first octant of an isotropic 3D harmonic oscillator.
I know the answer I should get is $$dV=\frac{1}{2}k^{2}dk$$.
What I currently have is $$dxdydz=dV$$ and $$k=x+y+z. But from that point on, I'm stuck. Any hints or reference material would be greatly appreciated. Thank you.

 

[vanhees71]

You have to tell us more context. I've no clue, what "the first octant of an isotropic 3D harmonic oscillator" should be and what it's good for.